Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

WM <invalid@no.org> wrote:

Am 27.03.2025 um 12:18 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

[ .... ]

For finite sets cardinality is a meaningful notion expressing the number
of elements. But I dislike it because it is often extended to cover
infinite sets where it does not describe the number of elements.

You have not proposed any other measure for the "number of elements" of
an infinite set.  At least, not in a mathematically useful way.

[ .... ]

So, to come back to my original example, the "substance" of {0, 4, 8,
12, 16, ...} wrt N is 1/4.

Yes.

The substance of {1, 3, 5, 7, 9, ...} wrt {0, 1/2, 1, 3/2, 2, 5/2, 3,
....} is also 1/4.

Yes.

Their "subtances" are thus the same.

Yes. Their amounts of substance, to be precise.

Or their cardinality, to be even more precise.

No, to be imprecise. Their numbers of elements differ but their
cardinality is the same.

So tell us, O wise one, how many elements are there in {1, 3, 5, 7, 9,
...}?  And how many elements in {0, 4, 8, 12, 16, ...}?  Which of these
two numbers is bigger, and why?

[ .... ]

Tend to yes, but not reaching it.

I thought you just said you had a degree in maths.  But you don't seem to
understand the process of limits (a bit like John Gabriel didn't when he
was still around).

0/oo = 0. 1/oo is smaller than every definable fraction.

More crank talk.  Ordinary arithmetic is not defined on infinity.  And
"smaller than every definable fraction" is zero.

Study surreal and hyperreal numbers which appear even in modern
mathematics.

[ .... ]

Look, you don't know much. Not even surreal and hyperreal numbers. Why
should I take your word on other topics as fact?>

For your information, I bought a copy of the first edition of Conway's
"On Numbers and Games" back in 1979 and read it, before surreal numbers
were even called that.  And in surreal numbers "smaller than every
definable fraction" is also zero.  There is no notion of "undefinable
number" in the surreals any more than there is in the reals or the
natural numbers.

[ .... ]

The potential infinite is a variable finite. Cantor's actual infinity
is not variable but fixed. (Therefore Hilbert's hotel is potential
infinity.)

Hilbert's hotel is infinite, not "variably finite".

Then its number of guests and of rooms could not change.

That's your intuition misleading you again.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).